Math Help

Hey... requested my professor to give me a few questions to enable me to improve my skills... am having problems solving them however and was hoping that you could guys could help me:

Q1) Use a calculator to evaluate the following powers. Round the results to five decimal placeS. Each of these powers has a rational exponent. Explain how you can use these powers to define 3^( sqrt of 2 ) which has an irrational exponent.

so basically the value of 3^(sqrt of 2 ) comes after 3^(1414213/1000000)... hence, you could find the log of 3^14/10 which is .6679697566, then probably do the following : 3^(.667..+.667...+.667....+.667.. +.667 +.667.. +.667).

EDIT: damn .. what i did above doesnt make sense.. im soo confused.!!! i do know that i can somehow use logs by finding the log of one value and then adding the solution multiple times to find the value of 3^ ( sqrt of 2 )

Surely, this is not just an exercise in using a calculator! And this isn't a "word problem" so I'm not sure what you meant by that first sentence. The point of the exercise appears to me to be: You have already defined exponentials for any rational power by am/n= (am)1/n= [itex]^n\sqrt{a^m}[/itex].
Now, how do you define exponentials for irrational numbers? Every irrational number is the limit of some sequence of rational numbers- that's exactly what you are doing when you say, for example, pi= 3.1415926.... 3, 3.1, 3.14, 3.141, 3.14159, 3.141592, 3.1415926,... is a sequence of rational numbers (because they are terminating decimals which could be written as a fraction exactly as you did [itex]\sqrt{2}[/itex])

DEFINING ax to be the limit of [itex]a^{r_n}[/itex] where rn is a sequence of numbers converging to x is just defining ax to be continuous.

You are right: "you could find the log of 3^14/10 which is .6679697566, then probably do the following : 3^(.667..+.667...+.667....+.667.. +.667 +.667.. +.667)."

doesn't make sense. Yes, 314/10 is, approximately, 0.66790696... but it makes no sense to talk about 3 to a sum of that. Are you confusing the exponent
14/10= 1.4 with the whole thing: 314/10?